Yet another – and simpler – generalization of Ricci curvature to metric measure spaces has been devised, based on the works of Bakry, Emery and Ledoux [1]}, [2]}.
More specifically, they consider manifolds with density, i.e. Riemannian manifolds \(M^n\) , additionally endowed, with a smooth, positive density function \(\Psi = \Psi (x)\) , that induces weighted \(n\) - and \((n-1)\) -volumes, e.g. in the classical cases \(n = 2\)  and \(n = 3\) , volume, area and length. More precisely, the volume, area and length elements \(dV, dA, ds\)  of the weighted manifold \((M^n,\Psi )\)  are given by:
\(dV = \Psi dV_0, dA = \Psi dA_0, ds = \Psi ds_0\,,\) 
